The decomposition of graphs is well known in the literature.
For example, decomposition can be based on the partition of a graph into edge-disjoint subgraphs.
In our case, in the two graph-decomposition theorems we contract nonempty subsets of the vertex set \(V\)  of the labelled acyclic directed multigraph \(G\) .
The contraction of a nonempty set \(X\subset V\)  leads to a graph \(G/X\)  where all the vertices of \(X\)  are replaced by one vertex \(\tilde{x}\)  and the arcs with both ends in \(X\)  are removed.
In the first theorem, we have nonempty sets \(X\subset V\)  and \(Y=V\backslash X\) , giving \(G/X\)  and \(G/Y\) .
In the second theorem, we have nonempty sets \(X_1\subset V,X_2\subset V\)  and \(Y=V\backslash X_1\cup X_2\)  giving \(G/X_1/X_2\)  and \(G/Y\) .
Then, together with additional constraints given in the theorems, we have that \(G\)  is isomorphic to the VRSP of \(G/X\)  and \(G/Y\)  in the first theorem and that \(G\)  is isomorphic to the VRSP of \(G/X_1/X_2\)  and \(G/Y\)  in the second theorem.
In this paper, we recall the definition of the VRSP and the two graph-decomposition theorems given in [1]} and we relax the requirements of these two graph-decomposition theorems.
For the first theorem, the requirement was that for the arcs that have one end in \(X\)  and the other end in \(Y\)  (the set of arcs \([X,Y]\) ) the label of each arc is distinct.
We relax this requirement in the following manner.
The set of all arcs of \([X,Y]\)  with the same label must arc-induce a complete bipartite graph.
For the second theorem, the requirement was that for the arcs that have one end in \(X_1\)  and the other end in \(Y\)  (the set of arcs \([X_1,Y]\) ), the arcs that have one end in \(Y\)  and the other end in \(X_2\)  (the set of arcs \([Y,X_2]\) ) and the arcs that have one end in \(X_1\)  and the other end in \(X_2\)  (the set of arcs \([X_1,X_2]\) ) the label of each arc is distinct.
We relax this requirement in the following manner.
The set of all arcs of \([X_1,Y]\)  with the same label must arc-induce a complete bipartite graph and the set of all arcs of \([Y,X_2]\)  with the same label must arc-induce a complete bipartite graph.
Furthermore, the only restriction on the labels of the arcs in \([X_1,X_2]\)  is that the arcs of \([X_1,X_2]\)  must not have a label identical to a label of any of the arcs of \(A(G)\backslash [X_1,X_2]\) .
